I believe yes, because if the bacterial sequence of the gene is not native in the genome then the splicing mechanism of the euakryotic system will be expressed.

Agreed. If it's a sequence that has sites resembling canonical splice donor/acceptor sequences, or spliceosome recruitment motifs, it might get aberrantly spliced.

I'd say (given how carefully orchestrated splicing is) that you'd have to be pretty unlucky to get a really strong splice-favouring motif, so I'd expect the more likely outcome to be a small amount of unwanted splicing, which (depending on your experiment) might not be too problematic.

Things you could do if you really think this is happening AND is happening at a detrimental level:

Identify where the splice is occurring (reverse transcribe, clone splice forms, sequence), and if possible, mutate out acceptor/donor motifs with site directed mutagenesis.

Add a strong, canonical intron of your own: basically introduce an artificial intron that the splicing machinery readily recognises, somewhere in your gene, so that rather than getting confused and splicing stuff it shouldn't, the spliceosome splices your inserted intron and calls it a day. It's actually not uncommon to find that transgenes carrying a deliberately introduced intron express much better than ones without (this is generally the reasoning for introducing it in the first place).

Thank you John and Christian!